Notes

The key idea (not to be underestimated) of this section is:

Tight frames are best understood via their Gramian.

Indeed, a tight frame \(\varPhi =(f_j)_{j=1}^n\) is determined up to unitary equivalence (and normalisation) by its Gramian \(P=P_\varPhi \), which is an orthogonal projection matrix. The columns of \(P_\varPhi \) give a (canonical) copy of \(\varPhi \), and so the kernel of \(P_\varPhi \) is the space of linear dependencies between the vectors in \(\varPhi \), i.e.,

Since \(P_\varPhi \) is determined by \(\ker (P_\varPhi )\), this observation allows the theory of tight frames to be extended to any finite dimensional vector space over a subfield of \(\mathbb {C}\) which is closed under conjugation (see Chapter 4).

Many notions of equivalence of tight frames appear in the literature (see [Bal99], [HL00], [GKK01], [Fic01], [HP04]). Here we use a descriptive terminology (from which all of these can be described). For finite tight frames viewed as sequences of vectors, unitary equivalence is the natural equivalence, and when viewed as (weighted) projections (fusion frames), projective unitary equivalence is natural. Unitary equivalence is determined by the Gramian (Corollary 2.1), and projective unitary unitary equivalence is determined by certain m-products (see Chapter 8).

It is implicit in the Definition 2.1 of a tight frame that \({\mathscr {H}}\) be separable, i.e., have a countable orthonormal basis. The theory extends, in the obvious way, to nonseparable spaces, with J now an uncountable index set. In these cases, it turns out that all tight frames for \({\mathscr {H}}\) (with nonzero vectors) have the same infinite cardinality, i.e., the Hilbert dimension of \({\mathscr {H}}\). By way of contrast, if \({\mathscr {H}}\) has finite dimension d, then there exist tight frames for \({\mathscr {H}}\) with any countable cardinality greater than or equal to d.

We will have good reason to consider representations such as (1.3), where the sum \(\sum _{j\in J}\) is replaced by a continuous sum (with respect to some measure). This generalisation (see Chapter 16) will be called a continuous tight frame, with the special case of Definition 2.1 referred to as a (discrete) tight frame.

The book [HKLW07] covers the material of this section. It has a section on frames in \(\mathbb {R}^2\) (for tight frames in \(\mathbb {R}^3\) see [Fic01]). The popular article [KC07a, KC07b] advocates the use of tight frames in a number of engineering applications. It outlines standard terminology for frames (resulting from an e-mail discussion within the frame community), which we adopt, except for our preference of normalised tight frame over Parseval frame. In this parlance a ENPTF is a equal-norm Parseval tight frame, and similarly.

2.4.

Orthogonal bases and tight frames.

(a) Show that an orthogonal basis \((f_j)_{j\in J}\) for \({\mathscr {H}}\) is a tight frame if and only if all its vectors have the same norm and that it is a normalised tight frame if and only if it is an orthonormal basis.

(b) Show that if \((f_j)\) is a normalised tight frame, then \(\Vert f_j\Vert \le 1\), \(\forall j\in J\), and

In particular, the only normalised tight frames whose vectors all have unit length are the orthonormal bases.

2.5.

Unitary images of tight frames.

(a) Show that the image of a tight frame \((f_j)_{j\in J}\) under a unitary map U is a tight frame with the same frame bound.

(b) Show that if \((f_j)\) is a finite normalised tight frame for \({\mathscr {H}}\), and T is a linear map for which \((Tf_j)\) is also, then T is a unitary map.

2.6.

Projections of normalised tight frames are normalised tight frames.

A linear map \(P:{\mathscr {H}}\rightarrow {\mathscr {H}}\) on a Hilbert space is an orthogonal projection if \(P^2=P\) and \(P^*=P\). Show that if \((f_j)_{j\in J}\) is a normalised tight frame for a Hilbert space \({\mathscr {K}}\) and P is an orthogonal projection onto a subspace \({\mathscr {H}}\subset {\mathscr {K}}\), then \((Pf_j)\) is a normalised tight frame for \({\mathscr {H}}\). (This is obvious in the context of Theorem 2.2.)

It is a coisometry if \(L^*:{\mathscr {K}}\rightarrow {\mathscr {H}}\) is an isometry, i.e., \(L^*\) is norm preserving.

Let \(\varPhi \) be a finite normalised tight frame for \({\mathscr {H}}\), and \(Q:{\mathscr {H}}\rightarrow {\mathscr {K}}\) be a linear map. Show that the following are equivalent

(a) Q is a partial isometry, i.e., its restriction to \((\ker Q)^\perp =\mathop {\mathrm{ran}}\nolimits (Q^*)\) is an isometry.

(b) \(QQ^*\) is an orthogonal projection.

(c) \(Q^*Q\) is an orthogonal projection.

(d) \(Q\varPhi \) is a normalised tight frame (for its span).

Remark. Since unitary maps and orthogonal projections are partial isometries, this generalises Exercises 2.5 and 2.6. It appears as a special case in Exer. 3.5.

2.8.

\(^\mathtt{m}\)If U is an \(n\times n\) unitary matrix (or a scalar multiple of one) with entries of constant modulus, then an equal-norm tight frame for \(\mathbb {F}^d\) is given by the columns of the \(d\times n\) submatrix obtained from it by selecting anyd of its rows.

(a) When \(\mathbb {F}=\mathbb {R}\), such U, with entries \(\pm 1\), are called Hadamard matrices. Use the matlab function hadamard(n) (defined for n,\({n\over 12}\) or \({n\over 20}\) a power of 2) to construct equal-norm tight frames of n vectors in \(\mathbb {R}^d\).

Remark: It can be shown that if a Hadamard matrix exists, then \(n=1,2\) or n is divisible by 4. The Hadamard conjecture is that there exists a Hadamard matrix of size \(n=4k\), for every k. The smallest open case (in 2010) is \(n=668\).

2.9.

(a) Show the vectors \((v_j)_{j=1}^n\) , \(v_j=(x_j, y_j)\in \mathbb {R}^2\) are a tight frame for \(\mathbb {R}^2\) if and only if the diagram vectors\(w_j := (x_j+i y_j)^2\in \mathbb {C}\) sum to zero (in \(\mathbb {C}\)).

(b) Show that two tight frames for \(\mathbb {R}^2\) are projectively unitarily equivalent if and only if their diagram vectors are scalar multiples of each other.

(c) Show that up to projective unitary equivalence the only equal-norm tight frame of three vectors for \(\mathbb {R}^2\) is three equally spaced unit vectors.

(d) Show that all unit-norm tight frames of four vectors for \(\mathbb {R}^2\) are the union of two orthonormal bases. This gives a one-parameter family of projectively unitarily inequivalent unit-norm tight frames of four vectors for \(\mathbb {R}^2\).

(e) Show the tight frames of five unit vectors for \(\mathbb {R}^2\) with diagram vectors

are projectively unitarily inequivalent, and that none is the union of an orthonormal basis and three equally spaced vectors.

2.10.

Projective unitary equivalence in\(\mathbb {R}^2\).

(a) For unit-norm tight frames of n vectors for \(\mathbb {R}^2\) show that the equivalence classes for projective unitary equivalence up to reordering are in 1–1 correspondence with convex n-gons with sides of unit length (given by a sum of diagram vectors).

(b) What do subsets of orthonormal vectors correspond to on the polygon?

(c) What is the n-gon corresponding to the tight frame for \(\mathbb {R}^2\) given by n equally spaced unit vectors?

(d) Does every finite tight frames for \(\mathbb {R}^2\) correspond to some convex polygon?

2.11.

The complex conjugate of \({\mathscr {H}}\) is the Hilbert space \(\overline{{\mathscr {H}}}\) of all formal complex conjugates with addition, scalar multiplication and inner product given by

(a) Show that the conjugation map\(C:{\mathscr {H}}\rightarrow \overline{{\mathscr {H}}}:v\mapsto \overline{v}\) is antilinear.

(b) Suppose that \(\varPhi =(f_j)\) is a sequence of vectors in \({\mathscr {H}}\) and \(\overline{\varPhi }:=(\overline{f_j})\subset \overline{{\mathscr {H}}}\). Show that the frame operator and Gramian satisfy

(c) Suppose that \({\mathscr {H}}=V\), with V a subspace of \(\mathbb {C}^d\). Show that \(\overline{{\mathscr {H}}}\) is isomorphic to the subspace \(\overline{V}:=\{\overline{v}:v\in V\}\) of \(\mathbb {C}^d\), where \(\overline{v}=\overline{(v_j)}:=(\overline{v_j})\).

2.12.

Show the frame operator S for a sequence of vectors \(f_1,\ldots , f_n\) satisfies:

2.13.

Trace formula. Show that if \((f_j)_{j\in J}\) is a finite normalised tight frame for \({\mathscr {H}}\) and \(L:{\mathscr {H}}\rightarrow {\mathscr {H}}\) is a linear transformation, then its trace is given by

2.17.

Let \(\varPhi =(f_j)_{j\in J}\) be a finite sequence of vectors in \({\mathscr {H}}\), where \(d=\dim ({\mathscr {H}})\), and \(V:=[f_j]_{j\in J}\). Show \(\varPhi \) is a normalised tight frame for \({\mathscr {H}}\) if and only if

2.20.

(a) Express unitary equivalence up to reordering in terms of the Gramian.

(b) Express projective unitary equivalence up to reordering in terms of the Gramian.

(c) Show that a necessary, but not sufficient, condition for normalised tight frames \((f_j)_{j\in J}\) and \((g_j)_{j\in K}\) to be projectively equivalent up to reordering is that there is a permutation \(\sigma :J\rightarrow K\) with \(|\langle g_{\sigma j},g_{\sigma k}\rangle |=|\langle f_j, f_k\rangle |\), \(\forall j, k\in J\). In particular, the multisets \(\{ |\langle f_j,f_k\rangle |\}_{j, k\in J}\) and \(\{ |\langle g_j,f_k\rangle |\}_{j, k\in J}\) must be equal.

2.21.

(a) Show that normalised tight frames \(\varPhi \) and \(\varPsi \) are projectively unitarily equivalent up to reordering if and only if their complements are.

(b) Show that all equal-norm tight frames of \(n=d+1\) vectors in \(\mathbb {F}^d\) are projectively unitarily equivalent, and hence are equiangular.

2.26.

M. A. Naĭmark’s theorem.

An orthogonal resolution of the identity for a Hilbert space \({\mathscr {H}}\) is a one-parameter family \((E_t)_{t\in \mathbb {R}}\) of orthogonal projections on \({\mathscr {H}}\), for which \(t\mapsto E_t\) is left continuous, and

A generalised resolution of the identity is a family \((F_t)_{t\in \mathbb {R}}\), for which the differences \(F_t-F_s\), \(s<t\) are bounded positive operators, \(t\mapsto F_t\) is left continuous, and

Naĭmark’s theorem (see, e.g., [AG63]) says that every generalised resolution of the identity for \({\mathscr {H}}\) is the orthogonal projection onto \({\mathscr {H}}\) of an orthogonal resolution of the identity for some larger Hilbert space \({\mathscr {K}}\supset {\mathscr {H}}\).

(a) Let \((f_j)_{j=1}^n\) be a finite normalised tight frame for which none of the vectors are zero. Show that a generalised resolution of the identity is given by

(b) By Naĭmark’s theorem, there is a Hilbert space \({\mathscr {K}}\supset {\mathscr {H}}\), and an orthogonal resolution of the identity \((E_t)\) for \({\mathscr {K}}\), such that \(F_t = P E_t\), where P is the orthogonal projection of \({\mathscr {K}}\) onto \({\mathscr {H}}\). Conclude that

2.30.

Matrices with respect to a normalised tight frame.

Normalised tight frames can be used to represent vectors and linear maps in much the same way as orthonormal bases. Suppose that \((f_j)_{j\in J}\) and \((g_k)_{k\in K}\) are finite normalised tight frames for \({\mathscr {H}}\) and \({\mathscr {K}}\), and let \(V=[f_j]_{j\in J}\), \(W=[g_k]_{k\in K}\). Then the coordinatesx of \(f\in {\mathscr {H}}\) with respect to \((f_j)\), and the matrixA representing a linear map \(L:{\mathscr {H}}\rightarrow {\mathscr {K}}\) with respect to \((f_j)\) and \((g_k)\) are

(c) Show that the composition of linear maps satisfies \([ML]=[M][L]\).

(d) Suppose \(L:{\mathscr {H}}\rightarrow {\mathscr {H}}\), and \(W=V\). Show that f is an eigenvector of L for the eigenvalue \(\lambda \) if and only if \(Ax=\lambda x\), i.e., eigenvectors of L correspond to the eigenvectors of A that are in the range of \(V^*\).

(e) Show L and A have the same singular values, and hence the same rank.